Steven K. answered • 03/08/21
Harvard/Yale Grad with 25+ years teaching and tutoring
The title of the problem tells us that these are two similar right triangles, If the triangles are similar, then the ratio of their corresponding sides will be equal. Thus 16/12 [ratio of the short sides] = (x+9)/18 [ratio of the hypotenuses]. Solving gives us (4*18/3)=x+9, so x=15 (and the hypotenuse of the larger similar right triangle = 15+9-24 To find the perimeter of either triangle, we need to know the length of the unmarked side. Using Pythagoras, for the larger triangle, 162+ s2=242 so s2=576-256=320. s=8√5 The perimeter of the larger triangle is 16+24+8√5=40+8√5.inches or 8(5+√5) For the smaller triangle, s=6√5 [using Pythagoras] and the perimeter = 12+18+6√5=30+6√5 or 6(5+√5). Notice that the perimeter of the larger triangle is equal to the perimeter of the smaller triangle multiplied by the same side ratio (16/12).
